28 research outputs found
Bipartite partial duals and circuits in medial graphs
It is well known that a plane graph is Eulerian if and only if its geometric
dual is bipartite. We extend this result to partial duals of plane graphs. We
then characterize all bipartite partial duals of a plane graph in terms of
oriented circuits in its medial graph.Comment: v2: minor changes. To appear in Combinatoric
Partial duals of plane graphs, separability and the graphs of knots
There is a well-known way to describe a link diagram as a (signed) plane
graph, called its Tait graph. This concept was recently extended, providing a
way to associate a set of embedded graphs (or ribbon graphs) to a link diagram.
While every plane graph arises as a Tait graph of a unique link diagram, not
every embedded graph represents a link diagram. Furthermore, although a Tait
graph describes a unique link diagram, the same embedded graph can represent
many different link diagrams. One is then led to ask which embedded graphs
represent link diagrams, and how link diagrams presented by the same embedded
graphs are related to one another. Here we answer these questions by
characterizing the class of embedded graphs that represent link diagrams, and
then using this characterization to find a move that relates all of the link
diagrams that are presented by the same set of embedded graphs.Comment: v2: major change
Partial duality of hypermaps
We introduce a collection of new operations on hypermaps, partial duality,
which include the classical Euler-Poincar\'e dualities as particular cases.
These operations generalize the partial duality for maps, or ribbon graphs,
recently discovered in a connection with knot theory. Partial duality is
different from previous studied operations of S. Wilson, G. Jones, L. James,
and A. Vince. Combinatorially hypermaps may be described in one of three ways:
as three involutions on the set of flags (-model), or as three
permutations on the set of half-edges (-model in orientable case), or
as edge 3-colored graphs. We express partial duality in each of these models.Comment: 19 pages, 16 figure
On the interplay between embedded graphs and delta-matroids
The mutually enriching relationship between graphs and matroids has motivated discoveries
in both fields. In this paper, we exploit the similar relationship between embedded graphs and
delta-matroids. There are well-known connections between geometric duals of plane graphs and
duals of matroids. We obtain analogous connections for various types of duality in the literature
for graphs in surfaces of higher genus and delta-matroids. Using this interplay, we establish a
rough structure theorem for delta-matroids that are twists of matroids, we translate Petrie duality
on ribbon graphs to loop complementation on delta-matroids, and we prove that ribbon graph
polynomials, such as the Penrose polynomial, the characteristic polynomial, and the transition
polynomial, are in fact delta-matroidal. We also express the Penrose polynomial as a sum of
characteristic polynomials